CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Unlimited centered reconstruction creates new extrema on a step

Statement refuted

An unlimited centered-slope reconstruction with the reconstructed upwind forward-Euler update is TVD and range-preserving for every scalar grid sequence whenever $∣ a ∣ Δ t / h \leq 1$ .

Facts & Assumptions

Given: Cells $I_{i} = [i h, (i + 1) h)$ , $h = a = 1$ , $Δ t = 1 / 2$ , initial averages $U_{i} = 1$ for $i \leq - 1$ and $U_{i} = 0$ for $i \geq 0$ , and unlimited centered increments $σ_{i} = (U_{i + 1} - U_{i - 1}) / 2$ .

[F1]

Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions: for $a > 0$ , the direct update uses $F_{i + 1 / 2} = U_{i} + σ_{i} / 2$ and $U_{i}^{n + 1} = U_{i}^{n} - (Δ t / h) (F_{i + 1 / 2} - F_{i - 1 / 2})$ .

Counterexample

1.1givenF1algebra

The initial sequence has one jump from $1$ to $0$ , so $TV (U^{0}) = 1$ . The centered increments are $σ_{- 1} = (U_{0} - U_{- 2}) / 2 = - 1 / 2$ and $σ_{0} = (U_{1} - U_{- 1}) / 2 = - 1 / 2$ ; all other increments vanish.

2.1F1step 1.1algebra

The relevant upwind fluxes are $F_{- 3 / 2} = 1$ , $F_{- 1 / 2} = 3 / 4$ , $F_{1 / 2} = - 1 / 4$ , and $F_{3 / 2} = 0$ . With $Δ t / h = 1 / 2$ , the update gives $U_{- 1}^{1} = 9 / 8$ , $U_{0}^{1} = 1 / 2$ , and $U_{1}^{1} = - 1 / 8$ , with the adjacent exterior states still $1$ and $0$ .

3.1step 1.1step 2.1algebra∎

The new variation is $∣ 9 / 8 - 1 ∣ + ∣ 1 / 2 - 9 / 8 ∣ + ∣ - 1 / 8 - 1 / 2 ∣ + ∣ 0 - (- 1 / 8) ∣ = 3 / 2 > 1$ . The method creates both an overshoot and an undershoot at a CFL number of one half, refuting the statement.

Depends on

Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions

Used by

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